# Trajectory & Sequence

There will be many numbers visited as we traverse from some number $$n$$ all the way down to 1. Consider these visited numbers in the order they appear, any slice of this list of numbers will be a trajectory. We have a handy notation for this as follows:

$$
n \to F^1(n) \to F^2(n) \to \ldots \to F^m(n)
$$

If the **trajectory** ends with 1, we call this to be a **sequence**.&#x20;

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Note that a sequence can continue indefinitely after 1, and repeat many 1s until finally terminating. For our convenience, a **sequence** will refer to a **trajectory** that ends with 1 and does not repeat.
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For example, the number 17 has the following sequence:

$$
17 \to 52 \to 26 \to 13 \to 40 \to 20 \to 10 \to 5 \to 16 \to 8 \to 4 \to 2 \to 1
$$

Any _slice_ of this could be considered a trajectory:

* $$52 \to 26 \to 13$$
* $$10 \to 5 \to 16$$
* $$13 \to 40$$
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## Reduced Trajectories

As we can see, trajectories can be rather lengthy; even with small starting numbers. We have already mentioned that looking at odd numbers suffice, and indeed they will make analyzing these trajectories a lot prettier too. The notation will be similar to before:

$$
n \xrightarrow{x_0} R_0(n) \xrightarrow{x_1} R(n_0) \xrightarrow{x_2} R(n_1) \xrightarrow{x_3} \ldots \xrightarrow{x_m} R(n_{m-1})
$$

Here, the notation works as follows:

* $$n_0 = R_0(n)$$ and $$x_0$$ is the power of two used within $$R_0$$.
* $$n_i = R(n_{i-1})$$ and $$x_i$$ is the power of two used within that $$R$$.

Likewise, if the last number in the **reduced trajectory** is 1 then this is called to be a **reduced sequence**.

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Looking again at number 17, its reduced sequence this time:

$$
17 \xrightarrow{0} 17 \xrightarrow{2} 13 \xrightarrow{3} 5 \xrightarrow{4} 1
$$

Note that 17 appears twice at the start, because it is already an odd number and $$R_0$$ will not do anything, thus $$x_0 = 0$$.
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